Desensitized filters

ABSTRACT

A method and system for the design and implementation of filters is presented in which the filter&#39;s transfer function can be provided with a significant insensitivity to the filter&#39;s tap coefficient values. A desensitized digital filter includes a first halfband filter and a second filter coupled in cascade between an input of the digital filter and the output of the digital filter. In embodiments, the first filter has the transfer function F(z)=K(1+z −1 )(1+z −1 ) wherein K≠0 is a scale factor. The digital filter may also interact with an up-sampler or a down-sampler. A desensitized Hilbert transformer includes an FIR filter having filter-tap coefficients whose absolute values equal the absolute values of the coefficients of an FIR filter F(z) for which the product (1+z −1 )F(z) is a halfband filter coupled in cascade with a second filter.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent application claims the benefit of Provisional PatentApplication No. 60/953,355, filed Aug. 1, 2007, entitled “DesensitizedHalfband Filters,” which is incorporated herein by reference in itsentirety.

FIELD OF THE INVENTION

The invention relates generally to digital filters.

BACKGROUND

Finite impulse response (FIR) filters are commonly used digital filters.An FIR filter has an impulse response that settles to zero in a finitenumber of sample periods. FIR filters are inherently stable because FIRfilters require no feedback and have their poles at the origin (withinthe unit circle of the complex z plane). However, all digital filters,including FIR filters, are sensitive to perturbations in the filter'stap coefficients.

A digital filter constructed as a cascade of two or more subfilters canpossess the capability of lowering the filter's sensitivity to thesefilter coefficient perturbations. This property is described in J. W.Adams and A. N. Willson, Jr., “A new approach to FIR digital filterswith fewer multipliers and reduced sensitivity,” IEEE Trans. CircuitsSyst., vol. CAS-30, pp. 277-283, May 1983 [referred to herein as“Adams”] which is herein incorporated by reference in its entirety.

In general, during the design of an FIR filter, the filter's length andtap coefficient's are selected to meet pre-defined characteristics thatare usually specified in terms of the filter's frequency responseH(e^(jω)). One goal of FIR filter design is to select filter taps andfilter length that minimize stopband ripple and passband ripple for thefilter's frequency response. (For a digital filter, the frequencyresponse function is the filter's transfer function H(z), evaluated onthe unit circle in the complex z-plane, i.e., z=e^(jω).) The Remezalgorithm is often used in FIR filter design to solve for the filter tapcoefficients that produce the desired equal ripple passband and stopbandbehavior.

In Adams, an efficient digital FIR lowpass filter was implemented as acascade of a multiplierless prefilter and an amplitude equalizer.Instead of building the conventional filter as dictated by the designtechnique used, a less costly implementation is utilized for theprefilter (e.g., all tap coefficients set to one with the first nulloccurring at the beginning of the stopband). The prefilter is not anoptimal filter by itself. However, the cascaded amplitude equalizerfixes the imperfections of the rough pre-filter. The resulting efficientFIR lowpass filter meets the required design constraints for the FIRfilter and has both a reduction in hardware implementation costs as wellas improved sensitivity. Roughly speaking, the lowered sensitivity ofthe filter's frequency response to perturbations of the amplitudeequalizer's coefficients results from the filtering action of theprefilter.

A common component in digital circuitry for communication systems is thehalfband filter. Halfband filters are often used in cooperation withup-samplers and down-samplers in multirate systems when a sampling-ratechange is required. Because of the requirements of a halfband filter,the type of prefilter used in Adams cannot implement a desensitizedhalfband filter.

What is therefore needed are new structures for implementingdesensitized halfband interpolation filters.

BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES

FIG. 1 depicts a frequency response of an exemplary halfband filter.

FIG. 2A depicts the passband detail and FIG. 2B depicts the stopbanddetail for the halfband filter of FIG. 1.

FIG. 3 depicts the coefficients of the smallest (shortest) halfbandfilter which occurs when N=1.

FIG. 4 depicts the coefficients of a halfband filter having length 7.

FIG. 5 depicts an exemplary structure for a direct implementation of theconventional halfband filter having length 7.

FIG. 6A depicts the mathematics associated with a conventionalup-sampler (also referred to as a data-rate expander); FIG. 6B depictsan up-sampler operation; FIG. 6C depicts a halfband anti-imaging filter;and FIG. 6D illustrates the mathematical relations for an up-samplersystem.

FIG. 7 illustrates the structure of a degree fourteen (15-tap) halfbandanti-imaging filter with a data-rate expander having components of thefilter moved prior to up-sampling component.

FIG. 8 depicts the structure of the degree fourteen halfband filter ofFIG. 7 in transposed form.

FIG. 9 depicts a desensitized implementation of a 15-tap halfbandanti-imaging filter along with a data-rate expander, according toembodiments of the present invention.

FIG. 10 depicts an alternative implementation of a desensitized 15-taphalfband filter, according to embodiments of the present invention.

FIG. 11 depicts the definition for a sample-and-hold up-sampler.

FIG. 12 depicts an alternate implementation of a 15-tap desensitizedhalfband anti-imaging filter, using sample-and-hold up-samplers,according to embodiments of the present invention.

FIG. 13 depicts a transposed implementation of a desensitized 15-taphalfband filter, according to embodiments of the present invention.

FIG. 14 depicts the frequency response of an exemplary conventional15-tap halfband filter.

FIG. 15 depicts the frequency response of an exemplary 15-tap halfbandfilter using 10-bit tap quantization.

FIG. 16 depicts sensitivity plots for a conventional 15-tap halfbandfilter and a desensitized 15-tap halfband filter, according toembodiments of the present invention.

FIG. 17 depicts an optimal implementation of an MCM tree for theconventional transposed-form 11-tap halfband filter structure.

FIG. 18 depicts the conventional transposed-form 11-tap halfband filterstructure.

FIG. 19 depicts an optimal implementation of an MCM tree for thedesensitized 11-tap halfband filter.

FIG. 20 depicts this desensitized transposed-form 11-tap halfband filterstructure, according to embodiments of the invention.

FIG. 21 depicts an optimal implementation of an MCM tree for aconventional 19-tap halfband filter structure.

FIG. 22 depicts an optimal implementation of an MCM tree for adesensitized 19-tap halfband filter structure.

FIG. 23 depicts an optimal implementation of an MCM tree for adesensitized 55-tap halfband filter structure.

FIG. 24 illustrates the SPT terms of the coefficients for a conventionalthird halfband filter (THF).

FIG. 25 illustrates the SPT coefficients of Desensitized-47, accordingto embodiments of the present invention.

FIGS. 26A and B depict direct and transposed forms for exemplary FIRfilters.

FIG. 27 depicts a graph of an array of B branch nodes and S sum nodesused in a proof.

FIG. 28 depicts a transpose of the MCM block of FIG. 19.

FIG. 29 depicts an implementation of a desensitized 11-tap halfbandfilter, according to embodiments of the present invention.

FIG. 30A depicts a desensitized version of an exemplary 11-tap halfbandfilter (F7 of Goodman), realized in transposed form, according toembodiments of the present invention.

FIG. 30B depicts a desensitized version of an exemplary 11-tap halfbandfilter (F7 of Goodman), realized in a direct form, according toembodiments of the present invention.

FIG. 30C depicts an interpolator using an exemplary 11-tap halfbandfilter (F7 of Goodman), desensitized and using fractional coefficientsimplemented as hard-wired rightward bit-shifts, according to embodimentsof the present invention.

FIG. 31 depicts an alternate implementation of a desensitized 19-taphalfband filter, according to embodiments of the present invention.

FIG. 32 depicts the general alignment of the passband and stopband in ahalfband filter.

FIG. 33 depicts the magnitude of the frequency response of a 3-taphalfband filter.

FIG. 34 shows the operation of a decimator that down-samples an inputsequence by a factor of two by omitting every other input sample.

FIG. 35 depicts an exemplary desensitized 11-tap halfband filterimplemented in combination with a decimator, according to embodiments ofthe present invention.

FIG. 36 depicts an exemplary desensitized 11-tap transposed-formhalfband filter implemented in combination with a decimator, accordingto embodiments of the present invention.

FIG. 37 depicts a pole/zero plot of an exemplary IIR halfband filter.

FIG. 38 depicts an exemplary computer system, according to embodimentsof the present invention.

The present invention will now be described with reference to theaccompanying drawings. In the drawings, like reference numbers generallyindicate identical, functionally similar, and/or structurally similarelements. The drawing in which an element first appears is indicated bythe leftmost digit(s) in the reference number.

DETAILED DESCRIPTION OF THE INVENTION

The following detailed description of the present invention refers tothe accompanying drawings that illustrate exemplary embodimentsconsistent with this invention. References in the specification to “oneembodiment,” “an embodiment,” “an example embodiment,” etc., indicatethat the embodiment described may include a particular feature,structure, or characteristic, but every embodiment may not necessarilyinclude the particular feature, structure, or characteristic. Moreover,such phrases are not necessarily referring to the same embodiment.

Further, when a particular feature, structure, or characteristic isdescribed in connection with an embodiment, it is submitted that it iswithin the knowledge of one skilled in the art to affect such feature,structure, or characteristic in connection with other embodimentswhether or not explicitly described.

Other embodiments are possible, and modifications may be made to theembodiments within the spirit and scope of the invention. Therefore, thedetailed description is not meant to limit the invention. Rather, thescope of the invention is defined by the appended claims.

I. Overview

While techniques for designing digital filters are well known, anentirely new method for the design and implementation of halfbandfilters is presented here, one in which the filter's transfer functioncan be provided with a significant insensitivity to the filter's tapcoefficient values. Such insensitivity can be exploited in thefilter-design process to yield halfband filters with reduced hardwarerequirements, which can lead to circuits having lower power consumption,higher operating speeds, and smaller IC area.

As described above, because of the requirements of a halfband filter,the type of prefilter used in Adams cannot implement a desensitizedhalfband filter. Fortunately, however, a modification can be made to thetype of prefilter that was employed in Adams and it can yield a suitablehalfband prefilter. If we consider a cascade of two of the simplestprefilters from Adams, i.e., let P(z)=(1+z⁻¹)(1+z⁻¹) then this P(z) is ahalfband filter. We have: P(z)=1+2z⁻¹+1z⁻². The resulting prefilter(with taps 1, 2, 1) can be implemented cheaply because no realmultiplication is necessary. Multiplication by two (or any positive ornegative integer power of two) can be implemented in binary with a shiftoperation. The design of desensitized halfband filters is described infurther detail in Section II, below.

Halfband Filters

A common component in digital circuitry for communication systems is thehalfband filter. This section provides a brief review of halfband filtercharacteristics and requirements. An FIR halfband filter is an FIRdigital filter whose transfer function is of the form

$\begin{matrix}{{H(z)} = {{z^{- N}\left( {h_{0} + {\sum\limits_{{k = 1},3,\;\ldots}^{N}{h_{k}\left( {z^{k} + z^{- k}} \right)}}} \right)}.}} & (1)\end{matrix}$

Thus, an FIR halfband filter is a symmetric (hence, linear phase) FIRfilter of length 2N+1, for some odd integer N, for which allcoefficients h_(k), where k is an even integer, have the value zero,except for the coefficient in the center h₀ which is nonzero. Typically,h₀ has the value ½, and the other nonzero h_(k) coefficients can haveany desired values.

Since the factor z^(−N) in equation (1) satisfies |z^(−N)|=1 for all zon the unit circle of the complex z plane, the design of a halfbandfilter is often carried out using the real-valued (albeit physicallyunrealizable, so called zero phase) transfer function

$\begin{matrix}{{H_{0}(z)} = {h_{0} + {\sum\limits_{{k = 1},3,\;\ldots}^{N}{h_{k}\left( {z^{k} + z^{- k}} \right)}}}} & (2)\end{matrix}$which can be written as

$\begin{matrix}{{H_{0}\left( {\mathbb{e}}^{j\;\omega} \right)} = {h_{0} + {\sum\limits_{{k = 1},3,\;\ldots}^{N}{2h_{k}\cos\; k\;\omega}}}} & (3)\end{matrix}$for all unit-magnitude complex values z, i.e., for z=e^(jω). FIR filtersare, in fact, usually designed by working with a zero-phase function ofthe equation (3) type, and the factor z^(−N) of equation (1) is thenincluded to create the physically realizable transfer function H(z), forwhich |H(e^(jω))|=|H₀(e^(jω))|.

The halfband filter constraints on the h_(k) having even k (i.e.,h_(k)=0 for all even k≠0) provide all halfband filters with the generalfeature that the real-valued zero-phase part of the halfband filtersatisfies H₀(e^(jπ/2))=½ and, furthermore, the function H₀(e^(jω))possesses odd symmetry about this ω=π/2 value. That is, we haveH ₀(e ^(j(−ω+π/2)))+H ₀(e ^(j(ω+π/2)))=1, for all ωε[0,π/2].  (4)Typically, the halfband filter's nonzero coefficients are chosen suchthat the relation H₀(e^(jω))≈1 holds in the passband (i.e., it holds forlow frequencies within the interval ωε[0, π/2]) and the odd-symmetryrelation (4) then causes H₀(e^(jω))≈0 in the stopband (i.e., it holdsfor high frequencies within the interval ωε[π/2, π]). FIG. 32 depictsthe general alignment of the passband and stopband in a halfband filter.

FIG. 1 depicts the magnitude of the frequency response 100 of anexemplary halfband filter, illustrating the general features describedabove. As depicted in FIG. 1, the transfer function H(e^(jω)) 102 isapproximately 1 in passband 110 and is approximately 0 in stopband 120.Additionally, as shown in FIG. 1, the function H(e^(jω)) possesses oddsymmetry about the point ω=π/2. FIG. 2A depicts the passband detail andFIG. 2B depicts the stopband detail for the halfband filter of FIG. 1.

As depicted in FIGS. 2A and 2B, one consequence of the odd-symmetry ofH₀(e^(jω)) about ω=π/2 is that the passband ripples 212 of H₀(e^(jω))are exactly the same as a 180-degree rotated (upside-down) version ofthe stopband ripples 222. Also, since cos kω≈1 for all ω sufficientlynear ω=0, it follows from equation (3) that the coefficient values of awell-designed halfband filter will tend to satisfy h₀+2(h₁+h₃+ . . .)≈1. In fact, if H₀(e^(j0))=1 thenh ₀+2(h ₁ +h ₃+ . . . )=1  (5)and this situation will also be accompanied by H₀(e^(jπ))=0 due to theodd-symmetry of H₀(e^(jω)) discussed above.

A halfband filter may also be scaled by an arbitrary value P (asillustrated in FIG. 1) and this scaling will cause the passband level tobe at the value P, not 1. Such scaling is of course accomplished bymultiplying all h_(k) coefficient values by P. In this case, therelation (5) can be written ash ₀=2(h ₁ +h ₃+ . . . )  (6)wherein the scaling value P need not appear explicitly, and this moregeneral constraint of course accommodates the unsealed relation (5) forwhich h₀=½.

FIG. 3 depicts the coefficients of the smallest (shortest) halfbandfilter which occurs when N=1. Therefore, equation (1) implies athree-tap filter having the transfer function:H(z)=z ⁻¹(h ₁ z ¹ +h ₀ +h ₁ z ⁻¹)

This case accommodates the above-mentioned halfband prefilter which,after scaling P(z) by ¼, becomes

${P(z)} = {\frac{1}{4} + {\frac{1}{2}z^{- 1}} + {\frac{1}{4}{z^{- 2}.}}}$

Here, it will follow that H₀(e^(jω))=½+½ cos ω (i.e., we set h₀=½ andh₁=¼). FIG. 33 depicts the magnitude of the frequency response for this3-tap filter.

Direct Implementation

According to equation (1), the second-smallest halfband filter will havelength 7, and will correspond to the N=3 case:H(z)=z ⁻³(h ₃ z ³ +h ₁ z ¹ +h ₀ +h ₁ z ⁻¹ +h ₃ z ⁻³)This filter will be described by tap coefficients h₀, h₁ and h₃, asillustrated in FIG. 4.

FIG. 5 depicts an exemplary structure 500 for a direct implementation ofthe conventional 7-tap halfband filter of FIG. 4. In structure 500, theinput samples are delayed by three two unit delays 510 a-c and by asingle one unit delay 510 d. The samples are further processed by aseries of adders 540 a-d and a series of multipliers 530 a-c. An addermay be any one of a number of known types of adders, e.g., acarry-ripple adder, a carry-save adder, etc. As will be understood toone of ordinary skill in the art, the number of adders employed may varywhen different kinds of adders are used. Unless otherwise indicated tothe contrary, the number of adders mentioned at various points herein isunderstood to refer to the number of carry-ripple adders.

The multipliers 530 a-c are commonly referred to as “taps” and thecoefficients of the multipliers (h₀, h₁ and h₃) are referred to as the“tap coefficients.” The structure of FIG. 5 takes advantage of thesymmetric character of the tap coefficients, around the center tap h₀(as is conventionally done for most linear phase filterimplementations), to reduce the required number of multipliers, in thiscase to three.

In FIG. 5, adder 540 a receives a present input sample and an inputsample that has been delayed by z⁻⁶. The output of adder 540 a ismultiplied by the coefficient, h₃, of multiplier 530 c. Similarly, adder540 b receives an input sample delayed by z⁻²⁻ and an input sample thathas been delayed by z⁻⁴. The output of adder 540 b is multiplied by thecoefficient, h₁, of multiplier 530 b. Finally, multiplier 530 a receivesan input sample delayed by z⁻³. The outputs of multipliers 530 a, b, andc are combined by adders 540 c and 540 d.

Halfband filters are often used in cooperation with up-samplers anddown-samplers in multirate systems when a sampling-rate change isrequired. FIG. 6A depicts the mathematics associated with a conventionalup-sampler (also referred to as a data-rate expander). In the up-samplerdepicted in FIG. 6A, the data rate is being expanded by 2, resulting inan output, Y(z)=X(z²). As illustrated in FIG. 6B, performing up-samplingby 2 requires the data-rate expander produce 2 samples for every oneinput sample. The conventional way of up-sampling by 2, shown in FIG.6B, produces a zero following every input sample.

One consequence of an up-sampling operation is the appearance of imagesof lower frequencies at higher frequencies in the up-sampled signal'sspectrum. As depicted in FIG. 6C, the up-sampled signal has a spectrum602 whose magnitude is mirror imaged about the point ω=π/2. In amultirate system where the sample rate is being doubled, a halfbandfilter is employed to eliminate these images of lower frequencies thatappear at higher frequencies in the up-sampled signal's spectrum.

FIG. 6D depicts the mathematical relations for an up-sampler system600D. Up-sampler system 600D includes a data-rate expander (up-sampler)610 and a filter 620. In embodiments, filter 620 is a lowpass halfbandfilter. The use of a lowpass halfband filter in an up-samplingenvironment is illustrated in FIG. 6C. The up-sampled signal 602 (X(z²))is run through a lowpass halfband filter 620, filtering out the unwantedimages (denoted as the hashed signal portion 604).

A consequence of implementing a filter after an up-sampling operation isthat the digital circuitry of the filter must operate at a higher datarate then if the circuitry is built prior to the up-sampling operation(e.g., at least twice the rate, for an up-sampling by 2 operation).Therefore, it is preferable to move components of the digital filter infront of the up-sampling operation. While it is evident that theup-sampling occurs before the filtering for such anti-imaging filters,it is possible to move some components of the filter back through theup-sampler so that they operate at the lower frequency.

Therefore, the designer wants the up-sampling component located moreforward so as to cause as much as possible of the filter hardware tooperate at the lower sample rate. FIG. 7 illustrates the structure of adegree-fourteen (15-tap) halfband anti-imaging filter with data-rateexpander 700 having components of the filter moved prior to theup-sampling component. Structure 700 includes a delay chain 710 having aseries of one-unit delays (z⁻¹) 712 a-g, a set of multipliers (taps) 730a-e having tap coefficients (h₀, h₁, h₃, h₅, and h₇), and a set ofadders 740 a-g, each prior to the up-sampling components 780 a, b.Structure 700 further includes a one unit delay (z⁻¹) 715 and an adder745 following the up-sampling components 780 a, b.

Notice that the z⁻² elements that would have appeared near the top ofFIG. 7 have become z⁻¹ elements as they now appear within thelower-sample-rate part of the system. Notice also that the up-samplerhas had to bifurcate, becoming two up-samplers 780 a, b, one on eachside of the single z⁻¹ element (which has been interchanged with the h₀multiplier of FIG. 5, so that the h₀ multiplication can be performed atthe lower sample rate).

As described above, when an up-sampling by 2 operation is performed, azero is produced after every input sample. Therefore, in filter 700,up-sampler 780 a is putting out a signal having zeros every othersample. Up-sampler 780 b is also putting out a signal having zeros everyother sample. Because the output of up-sampler 780 a is delayed by one,the zeros produced by up-sampler 780 a are not occurring at the sametime as the zeros produced by up-sampler 780 b when both data streamsarrive at adder 745. Because a zero is always being added to a non-zerosignal, the adder operation 745 can be replaced by an interleavingoperation. Consequently, the elements included in block 790 of FIG. 7could be replaced by a multiplexer.

Transposed Configurations

An FIR filter may also be implemented in an alternate transposeconfiguration. It is convenient to obtain the transposed-form filterfrom a direct-form filter by reversing all signal-flow directions. FIG.8 depicts the structure 800 of the degree-fourteen halfband filter withdata-rate expander of FIG. 7 in transposed form. Structure 800 can beconsidered as including two blocks, an F(z) block 860 and a G(z) block870. F(z) block 860 includes a series of one unit delays 812 a-g in adelay loop 810. The branch nodes in the delay loop 710 of FIG. 7 havebeen replaced by summation nodes 814 a-g in delay loop 810 of FIG. 8.F(z) block 860 also includes a subset of multipliers (taps) 830 b-e,having tap coefficients (h₁, h₃, h₅, and h₇). G(z) block 870 includesthe remaining multiplier (tap) 830 a having tap coefficient (h₀) andthree one-unit delays 875 a-c.

Notice that the transposed-form filter of FIG. 8 has all multipliers 830a-e simultaneously operating on the same input sample, unlike thedirect-form filter of FIG. 7. Because multipliers 830 a-e operate on thesame input sample, techniques can be used to simplify the structure ofthe five multipliers in FIG. 8. This can result in a reduction in thecost of implementation of the filter. However, notice also that extradelays 875 a-c (not present in the direct-form filter) are required inthe G(z) block shown in FIG. 8, where the presence of the twoup-samplers dictates a separate path for the h₀ multiplier's output.

II. Desensitized Halfband Filter

Factoring Out a Prefilter

As described above, the transfer function of any halfband filter isgoing to have a polynomial of a special structure having zeros for everyother tap except for the center tap. To implement a desensitizedhalfband filter as a prefilter-equalizer cascade, it is necessary thatthe transfer function of the overall halfband filter have the transferfunction of the prefilter as a factor. In embodiments, the second-orderprefilter P(z) mentioned above (P(z)=(1+z⁻¹)(1+z⁻¹)) can be used. Thissection describes the necessary constraints on a halfband filter inorder for its transfer function to possess the factor (1+z⁻¹).

This requirement is simply the requirement that the polynomial in z⁻¹that defines the halfband transfer function H(z) possess a zero at z=−1,i.e., at ω=π. Viewed this way, the requirement becomes simply thatH(e^(jω))=0. It therefore follows, from the above discussion, that theset of desensitized halfband filters must be restricted to the set whosecoefficients obey equation (6). Equivalently, again using the aboveinsights, our desensitized halfband filters must satisfy H(e^(j0))=1.These limitations are reasonably mild, in that any well-designedhalfband filter will possess coefficients that approximately satisfyequation (6), as discussed previously.

Having limited the halfband filters to ones whose transfer functionpossesses the factor (1+z⁻) it will then happen—automatically—that thehalfband filter will also have a second (1+z⁻¹) factor. An easy proof ofthis fact is to simply recall that the transfer function H(e^(jω)) isperiodic of period 2π and also that the halfband filter's zero-phasefunction (3) is symmetric about the point ω=π. This means, of course,that H₀(e^(jπ))=0 can only occur if the derivative of H₀ is also zero atω=π. That is, a double zero is necessarily present at ω=π whenever onezero exists at ω=π.

Accordingly, the above proves that the transfer function of an FIRhalfband filter will possess the factor P(z)=(1+z⁻¹)(1+z⁻¹) if and onlyif any of these three equivalent conditions holds:|H(e ^(jω))|=0 at ω=π  i)|H(e ^(jω))|=1 at ω=0  ii)h ₀=2(h ₁ +h ₃+ . . . ).  iii)[collectively referred to herein as “Theorem”]

One more insight, regarding Condition iii) of this Theorem is worthmentioning: It is well known that, in a system such as depicted in FIG.7, the part at the output side serves simply to interleave the twostreams of samples that enter the two up-samplers. When such a systemprocesses a nonzero DC signal—say x(n)=d≠0 for all n—then one up-sampleris being fed with the DC signal x₁(n)=h₀×d for all n, while the other isbeing fed the DC signal x₂(n)=2(h₁+h₃+ . . . )×d for all n. Hence,unless the conditions of the above Theorem hold, the system's outputwill not be simply a DC output; it will contain an oscillation at thesystem output's Nyquist frequency whose amplitude will be proportionalto the amount by which the values h₀ and 2(h₁+h₃+ . . . ) differ.Moreover, such an error will also be present when the system isprocessing any input signal having a nonzero DC component. For thisreason, it is apparent that Condition iii), i.e., the conditionexpressed in equation (6), is not only a mild constraint on a halfbandfilter's transfer function, it is likely a very desirable constraint inan up-sampling environment.

Assuming now that the halfband filter's transfer function possesses thefactor (1+z⁻¹), it can be factored out of the transfer function H(z).For the degree-fourteen transfer function of the FIG. 7 system, forexample:

H(z) = (1 + z⁻¹)[h₇ + (−h₇)z⁻¹ + (h₇ + h₅)z⁻² + (−h₇ − h₅)z⁻³ + (h₇ + h₅ + h₃)z⁻⁴ + (−h₇ − h₅ − h₃)z⁻⁵ + (h₇ + h₅ + h₃ + h₁)z⁻⁶ + (h₇ + h₅ + h₃ + h₁)z⁻⁷ + (−h₇ − h₅ − h₃)z⁻⁸ + (h₇ + h₅ + h₃)z⁻⁹ + (−h₇ − h₅)z⁻¹⁰ + (h₇ + h₅)z⁻¹¹ + (−h₇)z¹² + h₇z⁻¹³]which can be rewritten asH(z)=(1+z ⁻¹){[a ₀ z ⁻⁶ +a ₁(z ⁻⁴ −z ⁻⁸)+a ₂(z ⁻² −z ⁻¹⁰)+a ₃(1−z⁻¹²)]+z ⁻¹ [a ₀ z ⁻⁶ −a ₁(z ⁻⁴ −z ⁻⁸)−a ₂(z ⁻² −z ⁻¹⁰)−a ₃(1−z⁻¹²)]}  (7)wherea ₀ =h ₁ +h ₃ +h ₅ +h ₇a ₁ =h ₃ +h ₅ +h ₇a ₂ =h ₅ +h ₇a₃=h₇.  (8)

Notice that the halfband filter's center-tap value h₀ is not explicitlyused in the definition of the a_(k) values—its value being understood asredundant information, in that equation (6) is assumed to hold.

Using the factorization of equation (7), it is possible to create animplementation of the halfband filter of FIG. 7 in the “desensitized”manner. FIG. 9 depicts a desensitized implementation 900 of a 15-taphalfband anti-imaging filter along with a data-rate expander, accordingto embodiments of the present invention. Although implementation 900includes a data-rate expander, as would be immediately appreciated bypersons of skill in the art, and described further below, thedesensitized halfband filter may be used alone, in an up-samplingenvironment with a data-rate expander or in a down-sampling environmentwith a decimator.

Desensitized structure 900 includes a delay chain 910 having a series ofsix one-unit delays (z⁻¹) 912 a-f, a set of multipliers (taps) 935 a-dhaving coefficients (a₀, a₁, a₂, and a₃), and a set of adders 940 a-gprior to the up-sampling components 980 a, b (i.e., in the lowersample-rate section). Structure 900 further includes a one unit delay(z⁻¹) 915 and an adder 945 following the up-sampling components 980 a,b. The output of adder 945 is fed into the (1+z⁻¹) block 960.

Adder 940 d receives the present input sample and an input sampledelayed by z⁻⁶ (after multiplication by −1). The output of adder 940 dis multiplied by coefficient a₃ by multiplier 935 d. Adder 940 creceives an input sample delayed by z⁻¹ and an input sample delayed byz⁻⁵ (after multiplication by −1). The output of adder 940 c ismultiplied by coefficient a₂ by multiplier 935 c. Adder 940 b receivesan input sample delayed by z⁻² and an input sample delayed by z⁻⁴ (aftermultiplication by −1). The output of adder 940 b is multiplied bycoefficient a₁ by multiplier 935 b. Multiplier 935 a receives an inputsample delayed by z⁻³ and multiplies the input value by the coefficienta₀. As will be appreciated by one of ordinary skill in the art, the −1multiplications can be omitted by the use of subtractors in place of theadders: 940 d, 940 c, and 940 b.

The outputs of multipliers 935 b, c, and d are combined and added to theoutput of multiplier 935 a in adder 940 g and subtracted from the outputof multiplier 935 a in adder 940 a. The output of adder 940 a is fed asinput into up-sampler 980 a and the output of adder 940 g is fed asinput into up-sampler 980 b.

The structure of FIG. 9 bears some similarities to the conventionalhalfband structure depicted in FIG. 7. Both structures require twoup-samplers (780 a, b in FIGS. 7 and 980 a, b in FIG. 9) and bothinvolve one z⁻¹ delay element (715, 915) and one adder (745, 945)following the up-samplers to interleave the two up-sampled sequences.Both systems require the same number of adders (seven) on thelow-data-rate side, although many of them have effectively becomesubtractors in the new system of FIG. 9. The system of FIG. 9, however,requires an additional delay element and an adder to finish theprocessing with the (1+z⁻¹) block at the system's output. This extracost is likely offset by the savings that occur elsewhere in the newstructure of FIG. 9. At the block-diagram level, for example, while theconventional structure requires five scaling multipliers 730 a-e (havingthe coefficient values h₀, h₁, h₃, h₅, h₇) the new structure requiresonly four scaling multipliers 935 a-d (having the coefficient values a₀,a₁, a₂, a₃). Also, one fewer z⁻¹ delay element is required on thelow-data-rate side of the system of FIG. 9 (seven delays appear in theconventional system of FIG. 7, while the new system of FIG. 9 requiresjust six).

Although the above section derived the desensitized structure for a15-tap halfband filter, its realization and its mathematical descriptionare simple enough that a similar design procedure for halfband filtersof other lengths would be appreciated by persons of skill in the art.

Alternate Forms for the Desensitized Halfband Filter

Just as the conventional halfband filter can be implemented in atransposed form, it may be useful to find a transposed-formimplementation of the desensitized halfband filter. To accommodate thepresence of an up-sampler in the system, as noted above, thetransposed-form of a conventional halfband filter needs a few extradelay elements that are not present in the direct-form filter (i.e., thethree z⁻¹ blocks at the bottom of FIG. 8). The transposed-formdesensitized structure will also have this property. As a step in thedirection of obtaining the transposed-form desensitized halfband, analternate direct-form structure can be created, one that possesses theextra z⁻¹ blocks.

The right side of (7) can easily be rearranged such that the part insidethe curly brackets is written as a sum of terms that each have just asingle a_(k) factor, yielding:

H(z) = (1 + z⁻¹){a₀z⁻⁶(1 + z⁻¹) + a₁z⁻⁴(1 − z⁻⁴)(1 − z⁻¹) + a₂z⁻²(1 − z⁻⁸)(1 − z⁻¹) + a₃(1 − z⁻¹²)(1 − z⁻¹)}.This form of H(z) easily leads to the implementation shown in FIG. 10.

FIG. 10 depicts an alternative implementation 1000 of a desensitized15-tap halfband filter, according to embodiments of the presentinvention. The implementation of FIG. 10 shares certain substructureswith that of FIG. 9. However, unlike the structure of FIG. 9, FIG. 10does not include any rate-changing operations. That is, it has not hadany part of it moved through an up-sampler or down-sampler—hence itpossesses z⁻² delay blocks 1012 a-f instead of the z⁻¹ blocks 912 a-f ofFIG. 9. Alternative implementation 1000 also includes a block 1090including a one unit delay 1092, two adders 1094 a, b, and three twounit delays 1095 a, b, and c. Since the structure of FIG. 10 hasparallel paths that all converge on the output path, each ending in ana_(k) multiplier, it has an ideal form for converting into an efficienttransposed configuration.

In block 1090, adder 1094 a subtracts an input sample delayed by z⁻¹from the present input sample to produce X(z)(1−z⁻¹) which is fed as aninput into the delay block 1010 and as input into adder 1040 d. Adder1094 b adds an input sample delayed by z⁻¹ to the present input sampleto produce X(z)(1+z⁻¹) which is fed through the three delays 1095 a, b,and c.

One additional modification may be useful in desensitized systems. Itconcerns the “pushing back” of the system output's (1+z⁻¹) block (e.g.,block 960 in FIG. 9) to the point where it can be combined withup-samplers. For this purpose, the symbol shown in FIG. 11 for a“sample-and-hold up-sampler” is defined. Such an up-sampler simplyoutputs two consecutive sample values, each equal to the latest inputvalue received, rather than outputting the input value followed by azero, as in a conventional up-sampler—i.e., FIG. 6B. While thiscomponent eliminates the need for a separate z⁻¹ delay element at thehigh-data-rate side of the system, it creates a structure wherein oneadder at the high output sample rate is still required. That is, whilethe interleaving operation employed in the conventional FIG. 7 systemdoes not require that an actual addition be performed by the adder atthe output node (in effect, such an addition would add two values, oneof which is always zero), the sample-and-hold up-samplers of the newsystem insert no zero values between input samples.

FIG. 12 depicts an alternate implementation of a 15-tap desensitizedhalfband anti-imaging filter, using sample-and-hold up-samplers,according to embodiments of the present invention. In FIG. 12, the(1+z⁻¹) block 960 of FIG. 9 has been pushed back and combined withup-samplers 980 a and 980 b creating sample-and-hold up-samplers 1285 a,b. Because these sample-and-hold up-samplers insert no zero valuesbetween input samples, the addition appearing at the output in theredrawn FIG. 12 version of FIG. 9 must actually perform as an adder.Thus, what gets eliminated by use of the sample-and-hold up-sampler issimply the interleaving operation (i.e., one multiplexer).

FIG. 13 depicts a transposed implementation 1300 of a desensitized15-tap halfband filter with data-rate expander, according to embodimentsof the present invention. FIG. 13 shows the result of reversing allsignal flows in the FIG. 10 direct-form filter, thereby creating thetransposed-form desensitized halfband filter structure. Transposedimplementation 1300 includes a series of six one unit delays 1312 a-f ina loop 1310, a set of four multipliers (taps) 1330 a-d, having tapcoefficients (a₀, a₁, a₂, and a₃), a series of adders 1314 a-e in loop1310, and three one-unit delays 1375 a-c following multiplier 1330 a.The transposed implementation also includes two adders 1342 a, b, twosample-and-hold up-samplers 1385 a, b, a one unit delay 1315, and anadder 1345 which receives as input the output of sample-and-holdup-sampler 1385 a and the output delay 1315.

In FIG. 13, much of the transposed filter has been moved back through apreceding up-sampler, resulting in many z⁻² delay blocks becoming z⁻¹blocks. The above-mentioned sample-and-hold up-samplers have also beenemployed in this filter. It can be compared with the conventional 15-taptransposed-form halfband filter of FIG. 8.

Whether or not there is an appreciable hardware savings at the blockdiagram level of the desensitized structures (FIGS. 12 and 13 vs. FIGS.7 and 8) the true value of the new architecture resides in the loweredsensitivity of the filter's transfer function to variations in thetap-coefficient values of the multipliers. Such lowered sensitivity canbe exploited to reduce the cost of implementing these multipliers.

To demonstrate the lowered sensitivity, the operation of a commerciallyavailable conventional filter is compared to the operation of thedesensitized 15-tap halfband filter of FIG. 12. The selectedconventional filter is a 15-tap halfband filter employed in a commercialproduct by Analog Devices (hereinafter the “AD filter”), using the tapcoefficients (normalized to integer values): h₀=8192, h₁=4964, h₃=±1102,h₅=273, h₇=±39. As discussed in A. N. Willson, Jr., “Desensitizedhalfband interpolation filters” in Proc. Midwest Symp. Circuits Syst.,Montreal, August 2007, pp. 1034-1037 [Willson] which is incorporatedherein by reference in its entirety, the AD filter does satisfy theh₀=2(h₁+h₃+ . . . ) equation (6) constraint above.

When the coefficients of the AD filter, re-normalized such that h₀=0.5,are rounded to 13 fractional bits, the halfband filter has the transferfunction whose magnitude plot is shown in FIG. 14. This plot isindistinguishable from a plot of the ideal transfer function (i.e., oneusing double-precision floating-point coefficients and computations).Because circuitry using floating point numbers would usually be tooexpensive, fixed-point numbers are preferred; these fixed-pointcoefficients must be quantized to a finite number of bits. One commondesign goal is to reduce the number of bits in the coefficients whilestill meeting the filter specifications.

An identical plot to FIG. 14 is also obtained when the filter isimplemented as a desensitized halfband filter of the FIG. 12 type, whereits multiplier coefficients a₀, a₁, a₂, a₃ have been obtained from theh_(k) coefficients via equations (8) and then quantized to just tenfractional bits (as opposed to 13 fractional bits in the conventionalstructure). By way of comparison, if the conventional structure'scoefficients are rounded to ten bits, a significant deterioration(almost 20 dB) of the halfband filter's stopband (shown in FIG. 15)results. Clearly, therefore, the desensitized structure could be used toachieve a [1(⅘)( 10/13)]×100>38% reduction in coefficient storagerequirements for this filter (where the ⅘ factor includes the fact thatthe desensitized structure needs just four coefficients, and the 10/13factor relates to the shortened coefficient word length).

Another perspective on this situation can be obtained by computing thesensitivities of the transfer function to the coefficient values. Due tothe relatively large variations in transfer function magnitude acrossthe stopband, we use the normalized sensitivity function:

$S_{p}^{H} = {\frac{\partial\left( {\ln\; H} \right)}{\partial\left( {\ln\; p} \right)} = {\frac{\Delta\; H}{H} \times {\frac{\Delta\; p}{p}.}}}$Using such a (frequency dependent) measure of transfer functionsensitivity, the curves shown in FIG. 16 are obtained, each of whichillustrates the sensitivity of the conventional structure or that of thedesensitized structure with respect to one of its coefficients (p).

The top chart of FIG. 16 depicts the magnitude of the transfer functionof a conventional 15-tap halfband filter in the stopband. The middlechart of FIG. 16 plots the sensitivities of the 5 coefficients of thenonzero taps, h₀, h₁, h₃, h₅, and h₇, of the conventional structure. Asshown in FIG. 16, the coefficients (specifically h₀ and h₁) are in thehigh range of the chart between the nulls, and between the second andthird nulls the sensitivities of h₀ and h₁ are off the middle chart. Thebottom chart in FIG. 16 plots the coefficient sensitivities for thedesensitized structure. Notice that, at the frequencies where thetransfer function has a zero, all sensitivities are very large. Notice,however, that the sensitivities for the desensitized structure aregenerally significantly lower than those of the conventional structureat the frequencies lying between the transfer-function zeros.

Signed-Power-of-Two Halfband Filters

The desensitized structures described above can also be used to obtainrealizations of halfband filters that require less hardware than theconventional halfband structures require. To do this, the focus is movedto a deeper level, below the block-diagram level considered above,closer to the physical structure that will be built. The actualconstruction of the multipliers is considered and the loweredsensitivity of new desensitized structures is exploited to achievesavings in the building of these multipliers.

Much work has been reported in the literature on the efficientimplementation of tap-coefficient multipliers in digital filters, and insimilar systems. For additional information on efficient implementationof tap-coefficient multipliers in digital filters, see A. G. Dempsterand M. D. Macleod, “Use of minimum-adder multiplier blocks in FIRdigital filters,” IEEE Trans. Circuits Syst.-II, vol. 42, pp. 569-577,September 1995 [Dempster], O. Gustafsson, “A difference based addergraph heuristic for multiple constant multiplication problems,” in Proc.IEEE Int. Symp. Circuits Syst., New Orleans, May 2007, pp. 1097-1100[Gustafsson I], and O. Gustafsson, “Lower bounds for constantmultiplication problems,” IEEE Trans. Circuits Syst.-II, vol. 54, pp.974-978, November 2007 [Gustafsson II], each of which is incorporated byreference in its entirety.

It has been shown that a “multiplier block” can be created, a structurewherein an input value X is multiplied by a set of fixed integercoefficients also referred to as a “multiple constant multiplication”(MCM) block—and, by sharing some internal computations, very efficientrealizations of these simultaneous multiplications can be obtained.Indeed, Gustafsson I and II have demonstrated the capability ofobtaining optimal (e.g., minimal adder) implementations. In applyingthis work to the sensitivity problem described herein, an importantinsight is developed that when systems are made less sensitive to theirsystem parameter values, then this insensitivity can be exploited topermit an implementation whose computations with these parameters canrequire fewer adders.

This section describes these signed power of two (SPT) desensitizedfilters. First, the implementation of a version of a small halfbandfilter using the conventional halfband architecture is described. Next,the implementation of a small halfband filter using the desensitizedstructure is described and the results of both implementations arecompared. This section uses two examples taken from a classichalfband-filter paper by D. J. Goodman, “Nine digital filters fordecimation and interpolation,” IEEE Trans. Acoust., Speech, SignalProcessing, vol. ASSP-25, pp. 121-126, April 1977 [Goodman], which isincorporated herein by reference in its entirety.

FIG. 18 depicts a conventional transposed-form 11-tap halfband filterstructure with data-rate expander 1800. Filter structure 1800 includes adelay loop 1810 having five one unit delays 1812 a-e and five adders1814 a-d. Structure 1800 also includes four multipliers 1830 a-d (whichcan be considered a “multiplier block” for ease of discussion), twoup-samplers 1880 a, b, an adder 1845 and a one-unit delay 1815 coupledbetween adder 1845 and up-sampler 1880 b. Because filter structure 1800is in transposed form, the G(z) block 1870 includes multiplier 1830 aand two one-unit delays 1875 a, b.

Consider this 11-tap halfband filter (Goodman's F7 filter) having taps{h₅ 0 h₃ 0 h₁ h₀ h₁ 0 h₃ 0 h₅} whereh ₀=512, h ₁=302, h ₃=−53, h ₅=7.It is possible to implement this filter as a conventionaltransposed-form halfband filter, using a total of just four adders,along with hardwired bit-shifts, to get the necessary products of eachh_(k) coefficient times an input sample value X, as shown in FIG. 17.This implementation uses an algorithm due to Gustafsson Ito obtain theoptimal “multiplier block” implementation.

The complete 11-tap conventional halfband filter structure is shown inFIG. 18, where the output values for the multipliers h₀ h₁ h₃ and h₅ areobtained from the outputs of the FIG. 17 MCM block.

Next, the 11-tap desensitized version of the 11-tap filter withdata-rate expander is designed. The coefficients for the desensitizedhalfband structure are computed, using the formulae indicated inequations (8):a ₀ =h ₁ +h ₃ +h ₅=256a ₁ =h ₃ +h ₅=−46a ₂ =h ₅=7.  (9)A transposed-form desensitized structure is used to implement thisfilter. FIG. 20 depicts a desensitized transposed-form 11-tap halfbandfilter structure with data-rate expander 2000, according to embodimentsof the present invention. Structure 2000 includes a delay loop 2010having four one unit delays 2012 a-d and three adders 2014 a-c.Structure 2000 also includes three multipliers 2030 a-c (which can beconsidered a “multiplier block” 2035 for ease of discussion), twosample-and-hold up-samplers 2085 a, b, an adder 2045 and a one-unitdelay 2015 coupled between adder 2045 and sample-and-hold up-sampler2085 b. Because filter structure 2000 is in transposed form, the B(z)block 2070 includes two one-unit delays 2075 a, b.

It is possible to use Gustafsson's algorithm to obtain an optimalimplementation for multiplier block 2035 that contains just two adders,as shown in FIG. 19. As described further below, the use oftransposed-form structures is not actually necessary; that is, it ispossible—for both the conventional halfband and for the desensitizedhalfband—to use a direct-form structure and to have the same numbers ofadders employed as are required for the transposed-form structure.

Comparing the multiplier blocks implemented in these two implementations(depicted in FIGS. 17/18 and FIGS. 19/20), it can be concluded that theinsensitivity of the new structure has allowed the desensitizedimplementation to reduce the required number of adders from four (FIG.17) to two (FIG. 19). Similarly, by exploiting the new structure'sinsensitivity, it is possible to achieve at least a one-adder reductionin the implementation of each and every one of the filters given inGoodman.

In this regard, it is noted that all of the filters given in Goodmanobey the h₀=2(h₁+h₃+ . . . ) constraint—a situation mentioned byGoodman, although it is not apparent that the connections given in theTheorem above were presented by Goodman nor was the consequence that thefilters all possess the P(z) factor. Certainly Goodman does not mentionthe possibility of implementing the filters in a two-factor cascade,hence achieving the insensitivity.

An additional example from Goodman is considered. The 19-tap “basebandfilter” of Goodman requires the tap-coefficients:h ₀=238, h ₁=149, h ₃=−46, h ₅=22, h ₇=±12, h ₉=6.Gustafsson I's algorithm obtains the optimal six-adder block shown inFIG. 21 for implementing this conventional 19-tap filter. In contrast,the desensitized version of this 19-tap halfband filter requires onlythe following five taps:a ₀ =h ₁ +h ₃ +h ₅ +h ₇ +h ₉=119a ₁ =h ₃ +h ₅ +h ₇ +h ₉=−30a ₂ =h ₅ +h ₇ +h ₉=16a ₃ =h ₇ +h ₉=−6a ₄ =h ₉=6which can be implemented by the three-adder block shown in FIG. 22.Notice that all multiplier blocks are shown to implement only positiveconstants times the input X When a negative multiplier-value isrequired, a subtraction of the output is performed, rather than anaddition, as the block outputs are entered into the larger structure.

Another advantage offered by the desensitized filter structure is thatthe smaller multiplier blocks obtained for the desensitizedimplementations also tend to have a smaller maximum tree depth. Forexample, the eleven-tap conventional filter example required afour-adder block with a depth-three tree (as shown in FIG. 17), whileits desensitized counterpart required a two-adder depth-two adder tree(as shown in FIG. 19). In a further example, the 19-tap conventionalfilter example required a six-adder block with a depth-three tree (asshown in FIG. 21), while its desensitized counterpart required athree-adder depth-two adder tree (as shown in FIG. 22). The matter oftree depth is important because, if the tree-depth is too large, thiscan become a severe obstacle to obtaining a filter implementation thatoperates efficiently at a desired high data rate.

As described in Willson, another approach to the implementation of adesensitized halfband filter would be to factor the complete halfbandprefilter polynomial P(z)=(1+z⁻¹)(1+z⁻¹) out of the H(z) transferfunction. That is:H(z)=(1+z ⁻¹)(1+z ⁻¹)[q ₆ +q ₅ z ⁻¹ . . . +q ₅ z ⁻¹¹ +q ₆ z ⁻¹²].If this type of factorization is done for the 19-tap halfband filterjust considered, the following results:H(z)=(1+z ⁻¹)(1+z ⁻¹){[q ₈(1+z ⁻¹⁶)+q ₆(z ⁻² +z ⁻¹⁴)+q ₄(z ⁻⁴ +z ⁻¹²)+q₂(z ⁻⁶ +z ⁻¹⁰)+q ₀ z ⁻⁸ ]+z ⁻¹ [q ₇(1+z ⁻¹⁴)+q ₅(z ⁻² +z ⁻¹²)+q ₃(z ⁻⁴+z ⁻¹⁰)+q ₁(z ⁻⁶ +z ⁻⁸)]}.This yields the desensitized halfband filter shown in FIG. 31. Using thesame “baseband filter” coefficient values,h ₀=238, h ₁=149, h ₃=−46, h ₅=22, h ₇=−12, h ₉=6this leads (using equations like equations (2) of Willson) to thevalues:q ₈=6, q ₇=−12, q ₆=6, q ₅=0, q ₄=16, q ₃=−32, q ₂=2, q ₁=28, q ₀=91which, when built in two blocks ({q₀, q₂, q₄, q₆, q₈} and {q₁, q₃, q₅,q₇} and put into the “direct form,” require a total of 5 adders toimplement. But, while this is a savings from the 6 adders required bythis filter as a conventional halfband, it is not as good as thethree-adder block discussed above and shown in FIG. 22. Nonetheless, forother filters this factorization may prove advantageous and should beconsidered.

FIG. 31 depicts an alternate implementation of a desensitized 19-taphalfband filter structure 3100, according to embodiments of the presentinvention. Structure 3100 includes a delay loop 3110 having eight oneunit delays 3112 a-h. Structure 3100 also includes a (1+z⁻¹)(1+z—1)block 3102 coupled between the input and the delay loop 3110, ninemultipliers 3130 a-i, an adder 3145 and a one-unit delay 3115.

Implementing Larger Desensitized Signed-Power-of-Two Halfband Filters:

This section investigates the advantages of using the desensitizedhalfband filter structure for the implementation of larger halfbandfilters. The investigation begins with another example filter taken froma commercial product. The data sheet for the AD9776/AD9778/AD9779 byAnalog Devices describes a 55-tap halfband interpolation filter(referred to herein as “ADI-55”). This halfband filter provides at least87.5 dB attenuation over its stopband, the interval 0.6π≦ω≦π. As abenchmark, by using Gustafsson's algorithm, an optimal multiplier blockthat implements ADI-55 using 16 adders in an adder tree of depth six canbe obtained. However, ADI-55 does not satisfy equation (6) above. Hence,the investigation must start with a new reference filter that doessatisfy this condition, and then use it as the basis on which to designa desensitized counterpart.

Using linear programming techniques, another 55-tap halfband filter isdesigned (referred to herein as “ANW-55”). ANW-55 has slightly betterstopband attenuation then ADI-55, over the same stopband (the interval0.6π≦ω≦π), and it satisfies equation (6). ANW-55 requires animplementation complexity comparable to ADI-55, as illustrated in Table1 below.

TABLE 1 ADI-55 ANW-55 Desensitized-55 Stopband att. 87.5 dB 89 dB 89.1dB Adds required 16 17 13 tree depth 6 5 5

The ANW halfband filter is then used to design a desensitized halfbandfilter, “Desensitized-55,” which is also summarized in Table 1. Not onlyis the Desensitized-55 filter implemented with three fewer adders thanADI-55, it has better stopband attenuation (by more than 1.5 dB) and itstree depth is just five. The 13-adder tree for Desensitized-55—againobtained via Gustafsson's algorithm—is shown in FIG. 23. In FIG. 23, thedepth of each adder is indicated next to the adder. That is, adders 2340a and b are at depth 1; adders 2340 c, d, and e are at depth 2; adders2340 f, g, and h are at depth 3; adders 2340 i and j are at depth 4; andadders 2340 k, l, and m are at depth 4. Furthermore, Gustafsson'salgorithm can be used in an alternate mode that finds an adder-tree with15 adders, but having a maximum depth of just three. Situations canexist wherein one may prefer to pay a small price, such as using twoadditional adders, to obtain a reduced tree depth.

The actual design of ANW-55, and then Desensitized-55, was performed bya procedure which started with a 55-tap halfband filter whose stopbandattenuation exceeded the target 87.5 dB of ADI-55. This design wasperformed using linear programming. The taps of the filter ANW-55 werecomputed as floating-point real numbers. They were rounded to 25-bitvalues that met the requirements of equation (6). These were thenconverted via the formulae indicated by equations (8) into the tapvalues of the desensitized structure. Now, both sets of 25-bitcoefficients (for the conventional filter and the desensitized filter)were manipulated via an ad hoc procedure:

They were written in SPT form. Then, repeatedly, certain nonzero LSBswere deleted until a combination of remaining bits in the set ofcoefficients was found for which the stopband attenuation was suitable(i.e., better than that of ADI-55). Due to the structure'sinsensitivity, the set of coefficients that would ultimately yieldDesensitized-55 was found to permit the deletion of LSBs having largerbit-weights than those in the set of coefficients that would yieldANW-55. Thus, it was possible to remove more nonzero LSBs from thedesensitized coefficient set than could be removed from the set yieldingANW-55. When it was judged that the best coefficients had been found,the use of Gustafsson's algorithm confirmed that these fewer SPT termsin the Desensitized-55 design correlated well with the requiring offewer additions than in the traditional structure, ANW-55. Mostsignificantly, Desensitized-55 was superior to ADI-55 as indicated inTable 1. It was deemed coincidental that the stopband attenuation ofDesensitized-55 turned out to be slightly better than that of ANW-55.

A Further Large-Filter Example

This section considers one final reasonably large halfband filter whoseimplementation can be improved by employing the desensitized halfbandfilter structure. This is a 47-tap halfband filter whose design isreported in A. Y. Kwentus, Z. Jiang, and A. N. Willson, Jr.,“Application of filter sharpening to cascaded integrator-comb decimationfilters,” IEEE Trans. Signal Processing, vol. 45, pp. 457-467, February1997 [Kwentus], which is incorporated by reference in its entirety,where it is called a “Third Halfband Filter,” (referred to herein as“THF”). The minimum stopband attenuation of the THF is approximately73.1 dB.

As in the case of ADI-55, the THF tap coefficients did not satisfyequation (6). Thus, the design proceeded in a manner similar to thatdescribed above. The process began with an over-designed halfband filterwhose 47 tap coefficients did satisfy equation (6). Those coefficientswere quantized and transformed into the coefficients of a desensitizedhalfband structure using SPT coefficient representation. What appearedto be an optimal set of nonzero LSBs were found that could be omittedwhile still keeping the minimum stopband attenuation greater than 73.1dB. The result of this process was a 47-tap halfband filter“Desensitized-47” that achieves a minimum stopband attenuation ofapproximately 73.38 dB and that requires a depth-three adder tree havingjust ten adders.

This compares favorably to the 16 adders in a depth-five tree that,according to Gustafsson's algorithm, are required to realize thecoefficients of THF given in Kwentus. That is, the resultingdesensitized halfband filter architecture provides a (16−10)/16

37.5% reduction in adders. FIG. 24 illustrates the SPT terms of thecoefficients of THF (given in Kwentus) and FIG. 25 illustrates the SPTcoefficients of Desensitized-47, according to embodiments of the presentinvention. It is evident that fewer SPT terms are required forDesensitized-47 and that, generally, the bit-weights of the terms arelarger than those of THF—for example, among all taps of Desensitized-47there is only one SPT term having a weight as small as 2⁻¹⁵ while thetaps of THF employ three SPT terms with weight 2⁻¹⁵ as well as four SPTterms having weight 2⁻¹⁶. The need for smaller bit-weight terms in THFis surely a reflection of the structure's higher sensitivity to thecoefficient values.

The “trellis-search” type design techniques given in C-L. Chen and A. N.Willson, Jr., “A trellis search algorithm for the design of FIR filterswith signed-powers-of-two coefficients,” IEEE Trans. Circuits Syst.-II,vol. 46, pp. 29-39, January 1999 [Chen], which is incorporated herein byreference in its entirety, can also be extended to the new class ofdesensitized halfband filters in order to create an easier, moresystematic, and better method of performing the type of design discussedhere for the 47-tap and 55-tap examples. The design successes obtainedby the present ad hoc methods provide considerable encouragement that amore systematic process stemming from Chen is likely to yield excellenthalfband filter designs that exploit the advantages inherent within thedesensitized structure to obtain significant reductions in hardwarecomplexity. Other design techniques may also yield improved methods ofperforming the type of design discussed here, one example being M.Aldan, A. Yurdakul, and G. Dundar, “An algorithm for the design oflow-power hardware-efficient FIR filters,” IEEE Trans. Circuits Syst.-I,vol. 55, pp. 1536-1545, July 2008 [Aktan], which is incorporated hereinby reference in its entirety.

Converting Transposed-Form Halfband Filters into Direct-Form Filters

Both the direct and transposed forms of FIR filter implementation havetheir own relative advantages and disadvantages. For example, if adesign goal is the maximization of operating speed, then a direct-formfilter can exhibit the disadvantage of having a longer critical paththan that of its transposed-form counterpart. (As shown in FIG. 26, theformer must accomplish one multiplication and many successive additionsin its critical path, while the transposed-form filter has a criticalpath consisting of just one multiplication and one addition.) One theother hand, if the minimization of power consumption is a design goal,along with achieving a high operating speed, the use of carry-saveadders in the transposed-form filter can require additional powerconsumption due to a doubling of the number of registers needed toimplement the filter's delay chain (in comparison to the delay chain ofthe direct-form filter) because both carry and sum values would requiredelaying as each adder's output value is sent toward the next adder.

Thus, if possible, it is important for a designer to have the option ofchoosing either the direct form or the transposed form for the physicalimplementation of an FIR filter. Fortunately, it is known from O.Gustafsson and A. G. Dempster, “On the use of multiple constantmultiplication in polyphase FIR filters and filter banks,” in Proc.Nordic Signal Process. Symp., June 2004, pp. 53-56 [Gustafsson III],which is incorporated herein by reference in its entirety, that, whenimplementing an FIR filter that uses an MCM block architecture—e.g., byusing Gustafsson's algorithm—this implementation can be accomplished ineither the direct or transposed form while incurring the same overalladder expense. Hence, if an adder tree for a transposed-form FIR filteris obtained, it can be “operated backwards” and the resultingmulti-input/single-output block can be employed in a direct-form versionof the same filter. Moreover, even though the transposed block will beone in which adder nodes are transformed into branch nodes and viceversa, the total number of adders required in the direct-form filterwill exactly equal the total number of adders required in thetransposed-form filter. The term “total number” is used here because itis important to include all adders needed in the summing up of thevarious paths in both structures. For example, while the three adders(2640 a, b, and c) shown in the direct-form implementation of FIG. 26Awill be absorbed into its transposed MCM block, the three adders (2645a, b, and c) appearing in the transposed-form implementation of FIG. 26Bwill remain external to its MCM block. However, these three adders mustbe added to the number of adders appearing inside thetransposed-filter's MCM block to obtain the total number of adders forthe transposed-form filter.

As mentioned above, the result that the same adder expense occurs inboth direct- and transposed-form filters is not new. One can, however,easily establish it as follows:

We actually prove that if a graph contains only two-input adder nodesand two-output branch nodes, and if the graph has exactly one input nodeand exactly one output node, then the total number of branch nodesequals the total number of adder nodes.

Proof: Without loss of generality, the graph can be put into the form ofan array of B branch nodes and S sum nodes, as shown in FIG. 27. Thearray has B+2S input terminals at the bottom and 2B+S output terminalsat the top. The graph is constructed by interconnecting an output (top)terminal with an input (bottom) terminal. All terminals will beconnected this way (since there is no need for a one-output branch nodeor a one-input sum node) except for one input terminal and one outputterminal, and these will be the system input and system output. Let Cdenote the number of interconnecting wires. Then 2B+C=1 and B+2S−C=1.Subtracting both sides of these equations from each other yields B=S.Q.E.D. (A proof given in Gustafsson III treats a somewhat more generalresult.)

In converting an FIR filter into a graph for the purposes of thisdiscussion, the z⁻¹ delay blocks are simply “short-circuited”. Thisyields the desired one-input/one-output graph consisting of aninterconnection of the terminals of the B and S nodes. When the systemcontains data rate expanders, these essentially get short circuited too.It is, however, simpler to just move them back out of the filter, thenthe filter and its transpose can be compared to one another directly.

To illustrate the invariance of the number of adders when comparingdirect-form and transposed-form filters, or, equivalently, the overallequality in the number of branch nodes and sum nodes, consider theexample of the MCM tree of FIG. 19. When it is used in thetransposed-form system of FIG. 20, two of its outputs encounter a branchnode and then three such branches feed sum nodes. Hence, the completegraph of the system has 4+4=8 branch nodes (4 being internal to the MCMblock) and 2+6=8 sum nodes (2 being internal to the MCM block). Inparticular, there are 8 adders.

FIG. 28 depicts the transpose of the FIG. 19 multiplier block. Noticethat it is trivially constructed by replacing branch nodes in FIG. 19with sum nodes, and vice versa. The roles of input and output nodes arealso reversed. Clearly there are 4 sum nodes (2840 a-d) and 2 branchnodes (2850 a, b).

FIG. 29 depicts an implementation of a desensitized 11-tap filter 2900,according to embodiments of the present invention. Implementation 2900has a similar structure to the alternate implementation of thedesensitized 15-tap filter described above in FIG. 10. Implementation2900 also includes an input block 2990, a delay loop 2910, a multiplierblock 2935, adder 2940 a, adder 2940 b, and a (1+z⁻¹) block 2960.

Input block 2990 includes a one unit delay 2992, two adders 2994 a, b,and two two unit delays 2995 a and b. In block 2990, adder 2994 asubtracts an input sample delayed by z⁻¹ from the present input sampleto produce X(z)(1−z⁻¹) which is fed as an input into the delay loop 2910and as input into adder 2940 a. Adder 2994 b adds an input sample delayby z⁻¹ to the present input sample to produce X(z)(1+z⁻¹) which is fedthrough the two delays 2995 a and b.

Delay loop 2910 includes four two unit delays 2912 a-d. Multiplier block2935 includes three multipliers 2930 a, b, and c. The outputs ofmultipliers 2930 a, b, and c are combined by adders 2940 c and d. Theoutput of adder 2940 d is fed as an input to the (1+z⁻¹) block 2960.

When the block of FIG. 28 is used in the 11-tap direct-form desensitizedsystem of FIG. 29, the overall system has 4+4=8 sum nodes and 2+6=8branch nodes. Again, in particular, there are 8 adders.

Although it was not required for the present analysis, FIG. 28 shows thevalue of the block's output as a function of its inputs A, B, C.Clearly, it transposes the FIG. 19 Input/Output relation.

Important Observation Regarding the FIG. 9 System:

When transposing the MCM-block that is suitable for the transposed-formstructure of FIG. 13, to create a block that can be used in adirect-form realization, the block obtained is appropriate for the FIG.10 type structure. However, when the halfband filter is built as adirect-form filter, the FIG. 9 type architecture (or its FIG. 12variant) is preferable since it does not include the extra z⁻¹ blockslocated at the bottom of FIG. 10. (Moreover, if the direct-form filteris intended for use in an up-sampling system, the FIG. 10 filter is noteasy to “move back” through the up-sampler, due to the presence of thez⁻¹ block at the input.) Fortunately we are not necessarily stymiedhere.

When the center tap h₀ of the desired halfband filter equals ½, as itusually would, then the value of a₀ becomes ¼ (because of equations (6)and (8)). Thus, the a₀ tap's implementation becomes trivial; no addersare required. Hence, both of the FIG. 9 and FIG. 10 structures can bebuilt with the same number of adders and the MCM block that is suitablefor FIG. 10 is easily adapted for use in FIG. 9. FIG. 30( a) shows thedetailed realization 3000A of the transposed form FIG. 13 realizationand FIG. 30( b) shows the detailed realization 3000B of the direct formFIG. 12 realization. Notice that both FIG. 30( a) and FIG. 30( b) employa total of 8 adders each.

It is also useful to mention that, while tap coefficients have beennormalized to have integer values for most of the discussion herein, itis straightforward to re-scale them to have fractional values, as willlikely be preferable for practical implementation. The fractional tapcoefficients for the FIG. 30( b) circuit are the a_(k) values given inequations (9), but divided by 1024 (so that a₀=¼). That is, we want:a ₀=¼a ₁=− 46/1024a ₂= 7/1024.  (10)By starting at the left side of FIG. 28, and pushing a 10-bitright-shift operator through the transposed tree, any one of severalequivalent structures can be obtained that each implement the desiredre-scaled system and that only involve right-shifts. This process leadsto, for example, FIG. 30( c), which is a re-scaled version 3000C of FIG.30( b) that employs the fractional coefficients of equations (10).Halfband Decimation Filters

In addition to a halfband filter's use in combination with data-rateexpanders, halfband filters may also be used along with decimators insystems wherein an input signal's data rate is lowered. In such systemsthe input data must be lowpass filtered prior to the decimationoperation so as to avoid creating so-called aliasing components in theoutput data. FIG. 34 shows the operation of a decimator thatdown-samples an input sequence by a factor of two by omitting everyother input sample. FIG. 34A shows this operation in the time domain.Unfortunately, while the expected result is produced for thelow-frequency part of the input spectrum, the high-frequency part of theinput not only “gets through” the decimator, it gets moved (aliased)into the lower frequency spectrum as shown in FIG. 34C. Notice that themanner in which high frequencies are mapped into the lower frequencyrange is by a “folding” of the high frequencies around the ω=π/2 point,as illustrated in FIGS. 34B and C. (A frequency ω₁ within the interval(π/2, π] is mapped to the frequency π−ω₁.) To avoid this aliasing, wemust lowpass-filter the input signal before the decimation operation.This is shown in FIG. 34B, where a halfband filter is used.

FIG. 34D depicts a down-sampler system 3400 that includes a filter 3420and a decimator 3430. For down-sampling systems employing the halfbandfilter and decimator combination of FIG. 34D, it is well known to movefilter components forward, through the decimator, if possible, enablingthe operation of much of the filter hardware at the lower sample rate.For the desensitized halfband filter, the filters of FIGS. 10 and 29 areparticularly suitable for such halfband/decimator combinations. FIG. 35shows an example of such a combination that is based on the use of theFIG. 29 11-tap halfband filter and FIG. 36 shows a transposed-formrealization of the same halfband/decimator combination.

FIG. 35 depicts an implementation of a desensitized 11-tap filterimplemented in combination with a decimator 3500, according toembodiments of the present invention. Implementation 3500 includes aninput block 3590, a delay loop 3510, and a multiplier block 3535.

Input block 3590 includes a (1+z⁻¹) block 3565, two decimators 3595 a,b,a one unit delay 3592, two adders 3594 a, b, and two one-unit delays3575 a, b. The output of (1+z⁻¹) block 3565 is fed as input to decimator3595 a and as input to delay 3592. The output of delay 3592 is fed asinput to decimator 3595 b. In block 3590, adder 3594 a subtracts theoutput of decimator 3595 b from the output of decimator 3595 a. Adder3594 b adds the output of decimator 3595 b to the output of decimator3595 a.

Delay loop 3510 includes four two unit delays 3512 a-d. Multiplier block3535 includes three multipliers 3530 a, b, and c. The output of adder3540 a is fed as input to multiplier 3530 a and the output of adder 3540b is fed as input to multiplier 3530 b. The outputs of multipliers 3530a, b, and c are combined by adders 3540 c and d.

For systems implemented with desensitized halfband/decimatorcombinations, it is of interest to consider the implications of thepresence or absence of the equation (6) relation: h₀=2(h₁+h₃+ . . . ).By the Theorem of Section 3, this relation is equivalent to|H(e^(jω))|=0 at ω=π which, in fact, means that the halfband filter mustpossess a double zero at ω=π. A zero at ω=π will completely eradicateany spectral component of the X(e^(jω)) input signal at the input'sNyquist frequency, which is the frequency that the decimator wouldotherwise cause to become aliased directly onto the DC value of thespectrum of the output signal Y(e^(jω)). In fact, the presence of adouble zero at ω=π will tend to have a strong attenuating influence onall frequencies near the input's Nyquist frequency.

As is well known in the use of so-called CIC filters due to E. B.Hogenauer, “An economical class of digital filters for decimation andinterpolation,” IEEE Trans. Acoust., Speech, Signal Processing, vol.ASSP-29, pp. 155-162, April 1981 [Hogenauer], which is incorporated byreference in its entirety, such well-placed transmission zeros cansignificantly and efficiently protect low-frequency input signalcomponents in a down-sampling system wherein the information containedin these components is particularly important and must be protected fromcorruption by unwanted aliasing. Such situations may include oneswherein the halfband/decimator circuit is but the first stage of amulti-stage down-sampling operation. Thus, as in the up-samplingapplication, the equation (6) relation is quite likely one that isdesirable for reasons quite apart from its being required by ourdesensitized halfband filter structures.

Halfband-Like Filters

While the discussion thus far has focused on the design of halfbandfilters, there is another somewhat similar type of filter that canpotentially benefit from the present invention. It has been called a“halfband-like filter” in 0. Gustafsson, L. S. DeBrunner, V. DeBrunner,and H. Johansson, “On the design of sparse half-band like FIR filters,”in Proc. Asilomar Conf. Signals, Syst., Comp., 2007, pp. 1098-1102[Gustafsson IV], which is incorporated herein by reference in itsentirety. Such filters have transfer functions whose magnitude|H(e^(jω))| does not possess a halfband filter's precise odd-symmetryabout the ω=π/2 value, as discussed above in Section I. All halfbandfilters must have their passband and stopband aligned as shown in FIG.32. Notice that the passband ripple ±δ₁ must be exactly equal to thestopband ripple ±δ₂ and the passband edge ω_(p) and the stopband edgeω_(s) must be symmetrically located around ω=π/2. Halfband-like filtersdeviate somewhat from these requirements. One situation in which ahalfband-like filter could be superior to a halfband filter is where oneneeds a certain minimum amount of stopband attenuation, say 80 dB, butone does not need the accompanying extremely small passband ripple thatthe halfband filter's odd-symmetry dictates—in this case, approximately±0.0009 dB. A “halfband-like” filter can be designed, wherein thepassband ripple is specified to be, say, ±0.01 dB while the stopband isrequired to keep an 80-dB minimum amount of stopband attenuation. Thus,we have δ₁≈0.00115 while δ₂≈0.0001—which means that δ₁≈11.5×δ₂. Here thepassband and the stopband edges are positioned so as to maintain thesymmetric locations that they would have if the filter were a truehalfband filter. It is, of course, possible to envision a situationwherein the ω_(P)+ω_(S)=π halfband requirement is relaxed somewhat whilethe equality of passband and stopband ripples is maintained—or even asituation wherein both types of relaxation are performed while notdeviating excessively from a halfband filter's transfer function. Forall such halfband-like filters, it could be efficient to create thefilter in a manner wherein a prefilter is placed in cascade with anotherfilter, as in the halfband-filter designs that we have alreadyillustrated in detail. For example, a scaled version of the smallesthalfband filter P(z)=(1+z⁻¹)(1+z⁻¹) could be employed and thehalfband-like filter transfer function H(z) designed according to therelation H(z)=P(z)G(z).

Clearly, the cheaply built halfband filter P(z) can provide a coarseapproximation to the desired halfband-like transfer function, and thecascade implementation could inherit benefits due to the insensitivityof the overall transfer function H(z) to perturbations in the parametersof the transfer function G(z), as has been demonstrated above in thecase of halfband filters H(z). By deviating from the strict halfbandfrequency-domain requirements, it will happen that the halfband-likefilters will no longer have alternate tap multiplier coefficients ofvalue zero. Nonetheless, the enhanced insensitivity could still permitsome hardware savings to occur through efficient MCM tree savings and itmay even be possible to meet the less-demanding frequency-domain designconstraints with a lower degree transfer function H(z) than would bepossible when using a true halfband filter.

IIR Halfband Filters

In addition to the FIR halfband filters that have been discussed herein,it is well known that halfband filters can be built in IIR form. Themajor difference between FIR halfband filters and IIR halfband filtersis the fact that the former are described by a transfer function thatcan be put into the form of a polynomial in the variable z⁻¹ while thetransfer function of an IIR halfband filter is a rational function inz⁻¹. That is, in terms of the complex variable z, the transfer functionsof both an FIR and an IIR filter can be put into the form of a functionconstructed as a numerator polynomial in z divided by a denominatorpolynomial in z, where the roots of the numerator polynomial are thetransfer function's zeros and the roots of the denominator polynomialare the transfer function's poles. While system stability demands thatall poles lie within the unit circle of the z-plane, the distinguishingfeature of an FIR filter is that all of its poles will be located at theorigin, i.e., at z=0, while a stable IIR filter will have poles locatedat points other than z=0, although still within the unit-circle. Designmethods for IIR halfband filters can be found in S. K. Mitra, DigitalSignal Processing., 3^(rd) ed. New York: McGraw-Hill, 2006, Section13.6.5, pp. 787-790. [Mitra], which is incorporated herein by referencein its entirety, and in M. Renfors and T. Saramaki, “Recursive n-th banddigital filters, Parts I and II,” IEEE Trans. Circuits and Systems, vol.CAS-34, pp. 24-51, January 1987 [Renfors], which is incorporated byreference in its entirety, and elsewhere.

The presence of nonzero poles for an IIR filter does not interfere withthe possibility of organizing the transfer function in a way such thatit possesses at least two zeros at z=−1. Therefore, it is evident thatthe same kind of desensitizing advantages that have been exhibited forFIR halfband filters can be achieved as well by IIR halfband filters.

An example of an IIR filter design is presented in R. Yamashita, X.Zhang, T. Yoshikawa, and Y. Takei, “Design of IIR half-band filters witharbitrary flatness and its application to filter banks,” Electronics andCommunications in Japan, Part 3, vol. 87, No. 1, pp. 134-141, 2004[Yamashita], which is incorporated by reference in its entirety, whereinthe halfband filter's transfer function has the pole/zero plot shown inFIG. 37, where it is shown that the filter's transfer function has ninezeros at z=−1. If, in the building of such a filter, one or two of thesezeros were factored out of the numerator polynomial, and then theoverall transfer function was expressed in terms of the remainingnumerator polynomial coefficients (as in the FIR examples consideredabove using the a_(k) or q_(k) coefficients) this would clearly lead toexpressions for the transfer function value that would be less sensitiveto these parameters than would be the transfer function sensitivity tothe coefficients of the complete numerator polynomial.

Discrete-Time Hilbert Transformer

It is well known that a very simple relationship exists between FIRhalfband filters and discrete-time Hilbert transformers. In Mitra,Section 15.7, pp. 893-899. [Mitra2], which is incorporated herein byreference in its entirety, it is shown that a discrete-time Hilberttransformer H(e^(jω)) is related to a halfband filter G(e^(jω)) by arather simple relationship, namely:

${G\left( {\mathbb{e}}^{j\omega} \right)} = {{\frac{1}{2}{H\left( {\mathbb{e}}^{j{({\omega + {\pi/2}})}} \right)}} = \left\{ \begin{matrix}{1,} & {{{for}\mspace{14mu} 0} < {\omega } < \frac{\pi}{2}} \\{0,} & {{{for}\mspace{14mu}\frac{\pi}{2}} < {\omega } < {\pi.}}\end{matrix} \right.}$The relationship between the tap coefficients of G and H is shown in H.W. Schussler and P. Steffen, “Halfband filters and Hilberttransformers,” Circuits Systems Signal Processing, vol. 17, no. 2, pp.137-164, 1998 [Schussler], which is incorporated by reference in itsentirety. Given the halfband filter G, the Hilbert transformer H isobtained by getting the tap coefficients of H from:

${H(z)} = {{\sum\limits_{k = {- N}}^{N}{h_{k}z^{- k}}} = {2{\sum\limits_{n = {{- {\lfloor{N/2}\rfloor}} - 1}}^{\lfloor{N/2}\rfloor}{\left( {- 1} \right)^{n}g_{{2n} + 1}z^{- {({{2n} + 1})}}}}}}$which, specifically implies

$h_{k} = \left\{ \begin{matrix}{\mspace{85mu} 0} & {{{for}\mspace{14mu} k} = {2n}} \\{2\left( {- 1} \right)^{n}g_{k}} & {{{{for}\mspace{14mu} k} = {{2n} + 1}},{{{- \left\lfloor {N/2} \right\rfloor} - 1} \leq n \leq {\left\lfloor {N/2} \right\rfloor.}}}\end{matrix} \right.$These relations show that one can build a desensitized discrete-timeHilbert transformer by directly mapping the taps of a correspondingdesensitized halfband filter via the well-known relationships givenabove, and elsewhere.III. Exemplary Computer System

Embodiments of the invention may be implemented using hardware,programmable hardware (e.g., FGPA), software or a combination thereofand may be implemented in a computer system or other processing system.In fact, in one embodiment, the invention is directed toward a softwareand/or hardware embodiment in a computer system. An example computersystem 3802 is shown in FIG. 38. The computer system 3802 includes oneor more processors, such as processor 3804. The processor 3804 isconnected to a communication bus 3806. The invention can be implementedin various software embodiments that can operate in this examplecomputer system. After reading this description, it will become apparentto a person skilled in the relevant art how to implement the inventionusing other computer systems and/or computer architectures.

Computer system 3802 also includes a main memory 3808, preferably arandom access memory (RAM), and can also include a secondary memory orsecondary storage 3810. The secondary memory 3810 can include, forexample, a hard disk drive 3812 and a removable storage drive 3814,representing a floppy disk drive, a magnetic tape drive, an optical diskdrive, etc. The removable storage drive 3814 reads from and/or writes toa removable storage unit 3816 in a well known manner. Removable storageunit 3816, represents a floppy disk, magnetic tape, optical disk, etc.which is read by and written to by removable storage drive 3814. As willbe appreciated, the removable storage unit 3816 includes a computerusable storage medium having stored therein computer software and/ordata.

In alternative embodiments, secondary memory 3810 may include othersimilar means for allowing computer software and data to be loaded intocomputer system 3802. Such means can include, for example, a removablestorage unit 3820 and an storage interface 3818. Examples of such caninclude a program cartridge and cartridge interface (such as that foundin video game devices), a removable memory chip (such as an EPROM, orPROM) and associated socket, and other removable storage units 3820 andinterfaces 3818 which allow software and data to be transferred from theremovable storage unit 3820 to the computer system 3802.

Computer system 3802 can also include a communications interface 3822.Communications interface 3822 allows software and data to be transferredbetween computer system 3802 and external devices 3826. Examples ofcommunications interface 3822 can include a modem, a network interface(such as an Ethernet card), a communications port, a PCMCIA slot andcard, etc. Software and data transferred via communications interface3822 are in the form of signals, which can be electronic,electromagnetic, optical or other signals capable of being received bythe communications interface 3822. These signals are provided to thecommunications interface 3822 via a channel 3824. This channel 3824 canbe implemented using wire or cable, fiber optics, a phone line, acellular phone link, an RF link and other communications channels.

Computer system 3802 may also include well known peripherals 3803including a display monitor, a keyboard, printers and facsimile, and apointing device such a computer mouse, track ball, etc. In thisdocument, the terms “computer program medium” and “computer usablemedium” are used to generally refer to media such as the removablestorage devices 3816 and 3818, a hard disk installed in hard disk drive3812, and semiconductor memory devices including RAM and ROM. Thesecomputer program products are means for providing software (includingcomputer programs that embody the invention) and/or data to computersystem 3802.

Computer programs (also called computer control logic or computerprogram logic) are generally stored in main memory 3808 and/or secondarymemory 3810 and executed therefrom. Computer programs can also bereceived via communications interface 3822. Such computer programs, whenexecuted, enable the computer system 3802 to perform the features of thepresent invention as discussed herein. In particular, the computerprograms, when executed, enable the processor 3804 to perform thefeatures of the present invention. Accordingly, such computer programsrepresent controllers of the computer system 3802.

In an embodiment where the invention is implemented using software, thesoftware may be stored in a computer program product and loaded intocomputer system 3802 using removable storage drive 3814, hard drive 3812or communications interface 3822. The control logic (software), whenexecuted by the processor 3804, causes the processor 3804 to perform thefunctions of the invention as described herein.

In another embodiment, the invention is implemented primarily inhardware using, for example, hardware components such as applicationspecific integrated circuits (ASICs), stand alone processors, and/ordigital signal processors (DSPs). Implementation of the hardware statemachine so as to perform the functions described herein will be apparentto persons skilled in the relevant art(s). In embodiments, the inventioncan exist as software operating on these hardware platforms.

In yet another embodiment, the invention is implemented using acombination of both hardware and software.

CONCLUSION

While various embodiments of the present invention have been describedabove, it should be understood that they have been presented by way ofexample, and not limitation. It will be apparent to persons skilled inthe relevant art(s) that various changes in form and detail can be madetherein without departing from the spirit and scope of the invention.Thus the present invention should not be limited by any of theabove-described exemplary embodiments, but should be defined only inaccordance with the following claims and their equivalents.

What is claimed is:
 1. A digital filter comprising: a first filterincluding a delay loop coupled to a plurality of multipliers and havinga transfer function G(z) of degree greater than one; and a second filterimplemented in hardware and having a transfer function F(z)=K·(1+z⁻¹)where K≠0, wherein the first filter and the second filter are coupled incascade to make a filter having a transfer function H(z)=F(z)·G(z), withH(z) being a transfer function of a halfband filter.
 2. The digitalfilter of claim 1, wherein the first filter is a finite impulse response(FIR) filter.
 3. The digital filter of claim 1, wherein the secondfilter includes an additional (1+z⁻¹) block.
 4. The digital filter ofclaim 1, wherein the digital filter interacts with an up-sampler.
 5. Thedigital filter of claim 4, wherein the up-sampler is a sample-and-holdup-sampler.
 6. The digital filter of claim 1, wherein the digital filterinteracts with a down-sampler.
 7. The digital filter of claim 1, whereinthe first filter includes: a multiplier block coupled to the delay loop;and a plurality of adders coupled between the delay loop and themultiplier block.
 8. The digital filter of claim 7, wherein themultiplier block includes a plurality of filter multipliers, each havinga corresponding multiplier coefficient.
 9. The digital filter of claim8, wherein each of the multiplier coefficients is calculated using atleast one coefficient in a transfer function of the digital filter. 10.The digital filter of claim 7, wherein the multiplier block includes: aplurality of adders; and a plurality of shift elements.
 11. The digitalfilter of claim 10, wherein the multiplier block is configured togenerate a plurality of output values.
 12. The digital filter of claim1, wherein the digital filter is a finite impulse response (FIR) filter.13. The digital filter of claim 1, wherein the digital filter is aninfinite impulse response (IIR) filter.
 14. The digital filter of claim1, wherein the digital filter is a halfband-like filter.
 15. A Hilberttransformer, comprising: a first FIR filter including a delay loopcoupled to a plurality of multipliers and having a transfer functionG(z) of degree greater than one whose filter-tap coefficients haveabsolute values equal to sums of one or more filter-tap coefficients ofsaid Hilbert transformer, and a second filter implemented in hardwareand having a transfer function F(z)=K·(1+z⁻¹) where K≠0, wherein thefirst filter and the second filter are coupled in cascade to make afilter having a transfer function H(z)=F(z)·G(z), with H(z) being atransfer function of the Hilbert transformer.
 16. A digital filtercomprising: a first filter configured to provide a coarse approximationof a desired transfer function for the digital filter, the first filterincluding a delay loop coupled to a plurality of multipliers and havinga transfer function G(z) of degree greater than one; and a second filtercoupled to the first filter in cascade, wherein the second filter has atransfer function F(z)=K·(1+z⁻¹) where K≠0 and is configured tocompensate for the coarse approximation of the first filter, whereinH(z)=G(z)·F(z), with H(z) being a transfer function of a halfbandfilter.
 17. The digital filter of claim 16 wherein the first filter isan FIR filter.
 18. The digital filter of claim 16, wherein the secondfilter includes an additional (1+z⁻¹) block.
 19. The digital filter ofclaim 16, wherein the digital filter interacts with an up-sampler. 20.The digital filter of claim 19, wherein the up-sampler is asample-and-hold up-sampler.
 21. The digital filter of claim 16, whereinthe digital filter interacts with a down-sampler.
 22. A method forfiltering an input signal in a digital filter having a first filter,implemented in hardware, and a second filter, implemented in hardware,coupled in cascade, comprising: filtering the input signal in the firstfilter to produce a first filter output signal, wherein the first filterincludes a delay loop coupled to a plurality of multipliers and has atransfer function G(z) of degree greater than one; and filtering thefirst filter output signal in the second filter having a transferfunction F(z)=K·(1+z⁻¹) where K≠0 to produce an output signal of thedigital filter, wherein the digital filter has a transfer functionH(z)=F(z)·G(z), with H(z) being a halfband filter.
 23. The method ofclaim 22, wherein filtering the input signal in the first filterincludes filtering using an FIR filter.
 24. The method of claim 22,wherein filtering the first filter output signal includes filteringusing the second filter having an additional (1+z⁻¹) block.
 25. Themethod of claim 22, further comprising: receiving said input signal of adigital filter from an up-sampler.
 26. The method of claim 25, whereinthe up-sampler is a sample-and-hold up-sampler.
 27. The method of claim22, further comprising: down-sampling said digital filter output. 28.The method of claim 22, wherein filtering the input signal in the firstfilter includes producing a coarse approximation of a set ofcharacteristics of a desired output signal for the digital filter.
 29. Amethod for filtering an input signal in a Hilbert transformer,comprising: filtering the input signal in a first FIR filter including adelay loop coupled to a plurality of multipliers. and having a transferfunction G(z) of degree greater than one with filter-tap coefficientswhose absolute values equal sums of one or more filter-tap coefficientsof said Hilbert transformer; and filtering the first filter outputsignal in a second filter implemented in hardware and having transferfunction F(z)=K·(1+z⁻¹) where K≠0, such that G(z)·F(z) is the transferfunction of a Hilbert transformer.
 30. A method for filtering an inputsignal in a Hilbert transformer, comprising: filtering the input signalin a first filter implemented in hardware and having a transfer functionF(z)=K·(1+z⁻¹) where K≠0, to produce a first filter output signal; andfiltering the first filter output signal in an FIR implemented inhardware and including a delay loop coupled to a plurality ofmultipliers, and having a transfer function G(z) of degree greater thanone with filter-tap coefficients whose absolute values equal sums of oneor more filter-tap coefficients of said Hilbert transformer, such thatF(z)·G(z) is the transfer function of a Hilbert transformer.